I'm trying out the memoir class. My document so far:

\documentclass{memoir}
\usepackage{lipsum}
\chapterstyle{culver}

\title{My Title}
\author{My Author}

\begin{document}
\maketitle
\tableofcontents*

\chapter{Alpha}
\lipsum[1-2]

\chapter{Beta}
\lipsum[1-2]

\chapter{Gamma}
\lipsum[1-2]

\end{document}


I have a problem right now in that if I use the "culver" chapter style, the first page that includes the title page and table of contents ends up having the header. How do I get rid of this so that the title page doesn't include a header?

\thispagestyle{plain} is your friend here. Where it should go depends on the amount of the first chapter's text on the first page. Following your MWE this works:

...
\chapter{Alpha}
\lipsum[1-2]
\chapter{Beta}
\thispagestyle{plain}
\lipsum[1-2]
...


However, when the first chapter's text spills over to the following page use:

\chapter{Alpha}
\thispagestyle{plain}
\lipsum[1-3]
\chapter{Beta}
\lipsum[1-2]


Since the cluver chapter style sets the chapter page style to headings, it would be enough to use

\aliaspagestyle{chapter}{plain}


which is the standard in most cases.